# VCG

##### Example of Vickrey–Clarke–Groves (VCG) auction

Suppose two apples are being auctioned among three bidders.
• Bidder A wants one apple and bids £5 for that apple.
• Bidder B wants one apple and is willing to pay £2 for it.
• Bidder C wants two apples and is willing to pay £6 to have both of them but is uninterested in buying only one without the other.
First, the outcome of the auction is determined by maximizing bids: the apples go to bidder A and bidder B, since their combined bid of £5 + £2 = £7 is greater than the bid for two apples by bidder C who is willing to pay only £6. Thus, after the auction, the value achieved by bidder A is £5, by bidder B is £2, and by bidder C is £0 (since bidder C gets nothing).

##### Next, the formula for deciding payments gives:

• For bidder A: Bidders B and C have total value of £2 (the perceived value of the items they've won: £2 + £0). If A were removed, the maximizing bids would give C both the apples while B gets nothing. Hence, in this modified scenario, the value achieved by bidder B is £0 and by bidder C is £6. The total value achieved in this modified scenario by B and C is £6 (£0 + £6). So A pays £4 (£6 − £2).
• For bidder B: Bidders A and C have total value of £5 (£5 + £0). If B were removed, the maximizing bids would give both the apples to C while A gets nothing. Hence, in this modified scenario, the value achieved by bidder A is £0 and the value achieved by bidder C is £6, thus making the total value of £6 (£0 + £6). So B pays £1 (£6 − £5).
• Similarly, bidder C pays £0 ((£5 + £2) − (£5 + £2)).
After the auction, A is £1 better off than before (paying £4 to gain £5 of utility), B is £1 better off than before (paying £1 to gain £2 of utility), and C is neutral (having not won anything).